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In this lesson, we're going to be talking about more complicated structures where

Â rather than just having one or two, we now have several atoms per lattice point.

Â So we can begin with a structure that is of the type Mx2.

Â And when we look at a structure like this Mx2 what I have done is to illustrate one

Â of the ions as being blue and the other two as being black filled in circles.

Â And I've coupled them with dotted lines to indicate how those structures

Â are coupled together or how those ions are coupled together in the structure.

Â So when we look at this, what we're seeing here is again a face centered cubic

Â lattice and this time it has a total of three atoms per lattice point so

Â that gives us then a total of 12 ions in the unit cell.

Â And because of the stoichiometry,

Â what we have to have is two of the X type ions and

Â one of the M type so that the charge is balanced out in the unit cell.

Â We can look at the structure of a material which is based upon the Silica

Â tetrahedron, where we have silicons that sits in the center of the tetrahedron and

Â is surrounded by a total of four oxygen ions.

Â And one of the structures that can develop as a consequence of this tetrahedral

Â arrangement of silicon and oxygen is one of the forms of silica,

Â which is referred to a crystobalite.

Â There's several different crystal structures that are associated with SiO2

Â but this happens to be one of them and this one in particular

Â is a variation again on the face centered cubic structure.

Â So let's examine this in a bit more detail.

Â In terms of the FCC structure,

Â we're going to see that they're a total of six atoms per lattice point.

Â As we go through this, because we're dealing with the face centered cubic

Â structure, and we have six of the atoms associated with each lattice point,

Â we therefore must have a total of 24 of these atoms in the unit cell.

Â So let's take a look at this.

Â So here is our structure, and what I'm going to do is to identify the number

Â of the various types of species that are inside of the structure.

Â So the first thing we're looking at are all those indicated by

Â the number 1 and that 1 indicates the total number of silicon

Â atoms that are at those eight corner points.

Â Remember each one of those corner points has a total of eight unit cells around it,

Â so since we have eight corners, we there for

Â have one silicon atom that is associated with a corner.

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Now what we're going to do is we're going to take a look at

Â the silicon atoms that occupy the face centering positions.

Â So, when we look at those,

Â those are designated as 3s, meaning we have six faces.

Â Each one of the faces shares with an adjacent above or below,

Â in front or back and since we have six faces in every share we

Â have then a total of three silicon atoms that are associated with those faces.

Â Now we still have some silicons that are not labelled.

Â And we're going to go through and label those, and those there indicated as four.

Â And in this particular case when we look at the four, we see that there are a total

Â of four of those silicons that lie wholly inside of the unit cell.

Â And therefore, we must have a total of four silicon atoms that are inside and

Â associated with those particular atoms in space.

Â Now we're going to sum up all of the atoms that appear in the unit cell

Â in terms of the structure of silicon dioxide.

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Remember that we had a total of 8 silicon atoms and because the structure is

Â one to two, we therefore must have a total of 16 oxygen atoms per unit cell.

Â And as a consequence, we have a total of 24 atoms that

Â are in this unit cell of crystobalite, SiO2.

Â Let's look at another structure.

Â This happens to be once again a simple cubic structure.

Â And when you look at the structure, and

Â what I'll do is indicate what those various points mean

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and it has the stoichiometry of one titanium, one calcium and three oxygens.

Â And that is consistent with the picture that is up on the screen and

Â this particular structure, because of the importance and

Â the number of different compounds that crystallize in this form,

Â it's referred to as the Perovskite Structure.

Â What we can do is we can look at an alternative Perovskite structure of

Â titanium, calcium, oxygen material.

Â 6:01

And this time what we're going to do is reposition the atoms.

Â So the atoms now are located at the titanium at the corners.

Â And when we start looking at the calcium, it lies in the center.

Â And those two, then, are just simply interchanged and consequently,

Â we maintain the one to one relationship between titanium and calcium.

Â All right, now let's take a look at the oxygens.

Â Because of our new origin, the oxygens are not located on the faces,

Â but now they're located on the edges.

Â So let's do a little bit of counting here.

Â It turns out that we're dealing with a queue,

Â which means that we're going to have a total of 12 edges.

Â When we look at each one of those edges wherein oxygen is occupied.

Â Then what we find is,

Â each one of those oxygens is contributing a total of one quarter.

Â So hence, when we sum up all the edges and the one quarter contribution,

Â we'll get a total then of 12 divided by three or

Â four which is ultimately going to give us three.

Â So now we go back and we see the structure of calcium titanate.

Â Again, one titanium, one calcium and three oxygen.

Â So everything is consistent here.

Â If we were to go down the column of the periodic chart and replace calcium with

Â barium, we would come up with a structure barium titanate.

Â When we look at the structure of barium titanate,

Â it's similar to what we saw with calcium titanate with the exception that

Â the titanium is now displaced in the center of the unit cell.

Â As a result of that displacement, what we see is on the diagram to the right,

Â we see that the titanium has been moved up relative to the oxygen.

Â And then what we find is as a consequence of that charge displacement,

Â we develop a dipole that's associated with the structure now.

Â So we have a dipole.

Â And what does that mean?

Â Well, what happens in this particular case.

Â If we were to take this unit so and compress it along the C direction or

Â the Z direction, what we could do is to move the titanium back into

Â the center of the cell and the dipole then winds up the superion.

Â Well it turns out that there is a very fascinating material that can be

Â developed as a consequence of utilizing this structure, and that is we can produce

Â a material which can take electrical energy and mechanical energy,

Â or mechanical and electrical, and go back and forth between those two.

Â And this then becomes the basis for

Â something that we refer to as a transducer.

Â The last structure that I want to introduce in this lesson

Â is that associated with a crystalline polyethylene.

Â And remember we can begin to think about these polymer chains as spaghetti.

Â And what we've done in this particular case is to align each one of those

Â change in such a way that we produce this unit cell that happens to be orthogonal.

Â So the a, b and c have different dimensions but

Â again all of those interaxial angles are 90 degrees.

Â So you can begin to see as we move away from some of these simple structures and

Â we come in to these more complicated structures that we can still begin to

Â analyze the individual crystal structures of the various materials.

Â It just takes a lot more detail in order to do that.

Â Thank you.

Â